where \(\Phi \)  and \(\Phi ^\nu \)  are defined in (REF ) and (REF ), respectively. Clearly, \(\mathfrak {X} \subseteq {\mathcal {S}}^{\epsilon }\)  with \(\hat{\mathcal {S}}^{\epsilon } \cap \mathfrak {X} \ne \emptyset \)  and \( \mathfrak {X} \subseteq {\mathcal {S}}^{\epsilon ,\nu }\)  with \(\hat{\mathcal {S}}^{\epsilon ,\nu }\cap \mathfrak {X} \ne \emptyset \)  mean that an optimal solution of (REF ) is an optimal solution of (REF ), and an optimal solution of (REF ) is also an optimal solution of (REF ).
Therefore, according to Theorem 5.7 in [1]}, we can obtain that, under Assumption REF ,
\(&&\sqrt{\nu }(\Phi ^\nu (\hat{x}^{\epsilon ,\nu },\hat{u}^{\epsilon ,\nu })-\Phi (\hat{x}^\epsilon ,\hat{u}^\epsilon ))\\&&=\sqrt{\nu }(\frac{1}{\nu } \sum _{\ell =1}^\nu F(\hat{x}^{\epsilon ,\nu }(T),\xi _\ell )+\frac{1}{2}\Vert \hat{x}^{\epsilon ,\nu }-x_d\Vert _{L^2}^2+\frac{\delta }{2}\Vert \hat{u}^{\epsilon ,\nu }-u_d\Vert _{L^2}^2\\&& \quad \quad \quad - \mathbb {E}[F(\hat{x}^\epsilon (T),\xi )] - \frac{1}{2}\Vert \hat{x}^{\epsilon }-x_d\Vert _{L^2}^2-\frac{\delta }{2}\Vert \hat{u}^{\epsilon }-u_d\Vert _{L^2}^2)\\&&\xrightarrow{} \inf _{(x,u)\in \hat{\mathcal {S}}^{\epsilon } } Y(x,u),\) 
